Function approximation plays a pivotal role in AI, especially in machine learning, where the objective is often to approximate an unknown function (like a target variable as a function of input features) using a set of samples from the function. This approach underpins many machine learning algorithms, including neural networks, where the network learns to approximate a function mapping input data to outputs. It's essential for tasks where explicit forms of the function are not known but can be learned from data, such as in regression, classification, and reinforcement learning.

The concept of function approximation dates back to early statistical methods, but it gained significant prominence in the AI context with the development of neural networks in the 1980s. It is closely linked to the theory of universal approximation, which provides the theoretical basis for the ability of neural networks to approximate complex functions.

Key contributors to the theory and application of function approximation in AI include Kurt Hornik, Maxwell Stinchcombe, and Halbert White, who formalized the universal approximation theorem for neural networks in the late 1980s. This theorem is foundational in demonstrating that feedforward neural networks can approximate any Borel measurable function to any desired degree of accuracy, given sufficient complexity or size of the network.